Two identical particles of charge `q` each are connected by a masless spring of force cosntant `k` .They are placed over a smoothe horizontal surface .They are released wheen unstretched .If maximum extneison of the spring is `r`,the value of `k` is :(neglect gravitational effect)

Options

A. `k=(q)/(r)sqrt((1)/(piepsilon_(0)r))`
B. `k=(1)/(4piepsilon_(0))(q^(2))/(l^(2))xx(1)/(r)`
C. `k=(2q)/(r)sqrt((1)/(piepsilon_(0)r))`
D. `k=(q)/(r)sqrt((2)/(piepsilon_(0)r))`

Explanation

To solve the problem, we will analyze the forces acting on the two identical charged particles connected by a massless spring when they are released from an unstretched position.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two identical particles, each with charge \( q \), connected by a massless spring of spring constant \( k \).
- The particles are placed on a smooth horizontal surface and are released from an unstretched position.

2. **Identifying Forces**:
- When the particles are released, they will repel each other due to the electrostatic force (Coulomb's force) between them.
- The electrostatic force \( F_c \) acting on each particle is given by Coulomb's law:
\[ F_c = \frac{k_e \cdot q^2}{r^2} \]
where \( k_e = \frac{1}{4 \pi \epsilon_0} \) is Coulomb's constant, and \( r \) is the distance between the two charges when the spring is maximally extended.

3. **Maximum Extension of the Spring**:
- Let \( r \) be the maximum extension of the spring. At this point, the spring force \( F_s \) is equal to the electrostatic force \( F_c \).
- The spring force is given by Hooke's law:
\[ F_s = k \cdot r \]

4. **Setting Forces Equal**:
- At maximum extension, the spring force equals the electrostatic force:
\[ k \cdot r = \frac{k_e \cdot q^2}{r^2} \]

5. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ k = \frac{k_e \cdot q^2}{r^3} \]

6. **Substituting for Coulomb's Constant**:
- Substitute \( k_e = \frac{1}{4 \pi \epsilon_0} \):
\[ k = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{r^3} \]

7. **Final Expression**:
- Therefore, the value of the spring constant \( k \) in terms of the charges and the maximum extension is:
\[ k = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{r^3} \]

### Summary:
The final expression for the spring constant \( k \) is:
\[ k = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q^2}{r^3} \]